Learn Simplifying Fractions & Equivalent Fractions - [5-4-13] - Free Educational videos for Students in K-12

Learn Simplifying Fractions & Equivalent Fractions - [5-4-13] - By Math and Science

Transcript


00:00	Hello . Welcome back . This lesson is going to
00:02	be one of the most important that we're going to
00:04	have in this entire sequence . It's called simplifying fractions
00:07	. This is part one . Why is simplifying a
00:10	fraction ? So important . It's because later when we
00:12	add fractions or subtract fractions or multiply fractions or divide
00:16	fractions , we're going to get an answer and then
00:18	we're always going to be asked to simplify that fraction
00:21	to lowest terms or two simplest form . And that's
00:24	what we're doing here . We're practicing the skill of
00:26	simplifying a fraction because for literally every calculation we do
00:30	we'll have to simplify it at the end . So
00:32	here we have this lesson to practice that skill .
00:34	Now what does it mean to simplify a fraction .
00:36	All it means is you already know by now that
00:38	fractions can look different but actually they can mean the
00:41	same thing even though they look different . We talked
00:44	about equivalent fractions . We said we can multiply a
00:46	fraction by anything we want on the top and the
00:49	bottom . As long as we multiply top and bottom
00:52	by the same number . Then we get a new
00:54	fraction that's equivalent to the old one , even though
00:56	it looks different and the same thing with simplifying fractions
00:59	, it turns out you can also divide the top
01:02	of a fraction and the bottom of the fraction by
01:04	the same number . As long as you do it
01:06	to the same number , you don't change anything ,
01:07	you just change the way it looks . So the
01:10	big picture is that for fractions you can multiply the
01:13	fraction by anything you want top and bottom . And
01:15	you can also divide a fraction by anything you want
01:18	top and bottom . And when you do that you
01:20	get new fractions that look different , but they are
01:23	exactly equivalent . Simplifying fractions means we divide top and
01:27	bottom by something to get it into the simplest form
01:30	, but it means the exact same thing . So
01:32	let's take a look at that And see if we
01:34	can understand the concept here . What about the fraction
01:37	? 3/6 ? And we want to do what we
01:41	call simplifying that fraction now , first of all ,
01:43	what is 36 ? Right . Let's take a look
01:46	at what that actually is . Here's 162636 This is
01:53	36 of a fraction . Now , let me ask
01:55	you something if I turn it maybe this way ,
01:57	it might help see it a little better . What
01:59	does that actually look like to you ? Well 36162636
02:03	Here it is . But what does it look like
02:05	? It looks exactly like half of a pizza .
02:07	In fact , I can hold up this is the
02:09	fraction one half . I can hold up the fraction
02:11	one half and they look and they are exactly the
02:14	same thing . So we can see from these magnets
02:16	That the fraction one half of a pizza is exactly
02:19	the same as 36 . Why is that by the
02:21	way ? Because if you have a pizza that you
02:23	cut into six pieces , but I take three of
02:25	the pieces , then I've taken half of the pizza
02:28	, so we have to have a way or we
02:30	want to have a way to take any fraction that
02:33	we have and simplify it into a simpler version of
02:36	itself . So let me move this down below right
02:39	here and let's try to do that the way you
02:41	do it is the following . Put a little equal
02:43	sign here . Rewrite the fraction 3/6 . Now ,
02:47	is there a number that you can think of ,
02:49	where you can divide the top and the bottom and
02:52	it would go evenly . Could you divide by two
02:54	for instance ? Well six divided by two is goes
02:57	a whole number of times , but what is three
02:59	divided by two ? Three divided by two does not
03:01	divide evenly . So we can't divide by two because
03:04	we can't do it to the top and to the
03:05	bottom . What about the number three ? What if
03:08	we choose to divide the top of this fraction by
03:10	the number three ? And also the bottom of the
03:12	fraction by the number three ? What would happen ?
03:14	Well three divided by three on the top is just
03:17	one And six divided by three on the bottom is
03:21	just too . So what we have said here is
03:24	that the fraction one half here , which is represented
03:27	here is exactly the same thing as the fraction 3/6
03:30	. And of course you can see from these that
03:31	this is true . So let me rotate this ,
03:34	just kind of get them out of the way a
03:35	little bit more and we'll leave it up on the
03:37	board . What you would say is that this is
03:39	the simplified form or the lowest terms form of the
03:43	original fraction . It's important for you to understand that
03:46	the fractions mean the same thing . 36 really does
03:50	mean exactly the same amount of stuff is one half
03:53	. But when we give answers and problems , we
03:56	usually want to show the simplified version of the fraction
03:59	into lowest terms . Like in general when we solve
04:01	problems , we always want to simplify as the final
04:04	step to get the fraction into lowest terms into simplest
04:07	form . All right . So now that we have
04:10	the idea down , let's go conquer the next one
04:13	. Let's say we have the fraction 4/12 . And
04:17	I'm gonna ask you , is this in lowest terms
04:19	? Or can we simplify this into lowest terms ?
04:22	Well , what you need to be asking yourself is
04:24	can I divide the top and bottom by something ?
04:26	Anything I can think of and the answer is yes
04:29	. I noticed that this is a whole number .
04:31	Uh an even number . I mean this is a
04:32	four and this is a 12 . So actually I
04:35	can take this 4/12 right ? And I can because
04:40	I know that they're both even numbers , I can
04:41	divide the top by two and I can divide the
04:44	bottom by to write , what will I get ?
04:46	I'll get four divided by two is two and 12
04:50	divided by two is six . So I'll get a
04:52	new fraction . And we're saying that the fraction to
04:54	sixth is exactly the same thing as the fraction 4/12
04:58	. But then I look at this and I say
05:00	well wait a minute , is this in simplest form
05:02	? And I say no , it's not because I
05:04	could still divide the top again by two and I
05:07	could divide the bottom again by two . So let's
05:09	do that to sixth . Let's see what happens when
05:13	I divide the top by two and the bottom by
05:18	to what do I get Right ? two divided by
05:20	two is on the top uh one and six divided
05:24	by two on the bottom is three . And so
05:27	I get an answer of one third . So what
05:30	we're saying , and then we ask ourselves , is
05:32	this the final answer ? Is this a simplified form
05:34	, one divided by three or 1/3 ? I cannot
05:37	simplify this further because I cannot divide top and bottom
05:40	by something . To make it simpler . I can't
05:43	do it because even if I divide by three ,
05:45	that would work down here , but I can't divide
05:46	this by three , it doesn't go evenly . So
05:48	I'm done . I stop at the fraction . 1
05:50	3rd . All right , So let's see first of
05:53	all if this is even true , let's take a
05:55	look at the fraction . 4/12 here , here's 1/12
05:59	here's to 12th , here's 3/12 here's 4 12 .
06:03	So this is the actual amount of pizza . Notice
06:05	the slices when you cut a pizza into 12 pieces
06:08	pretty small . 4/12 . Now let's go all the
06:11	way to the answer . We said the answer was
06:13	one third were saying that these two fractions , even
06:16	though they look totally different , actually represent the same
06:19	thing . One third is exactly the same thing as
06:22	4/12 so that's why they're equivalent all of the numbers
06:25	in the fraction looked totally different , but the actual
06:27	fraction means the same thing . Now notice that first
06:32	we divided by two to get this and then we
06:34	simplified further in a second step to get this ,
06:37	let's see what this is equal to what is to
06:39	sixth here . Let me go over here to six
06:42	, here's 1/6 here's to sixth because these are now
06:45	cut into six larger slices , there's 26 and I
06:49	think you can see that one third is exactly the
06:51	same as to six , which is also exactly the
06:54	same as 4 , 12 . So you see what's
06:56	going on here . When I divide this number fraction
06:59	by two on the top and two on the bottom
07:01	I get this , this fraction means exactly the same
07:03	as this , but it's not fully simplified yet .
07:06	But when we divide by two again we get to
07:08	this which again means exactly the same amount of pizza
07:11	. But this is fully simplified because I cannot further
07:14	divide this anymore . So when you see a teacher
07:18	or a test tell you simplify the fraction into lowest
07:21	terms you want to keep simplifying as far or as
07:24	much as you need to in order to get to
07:26	a point where you can't do it anymore , where
07:28	you can't do it anymore . Now , one more
07:30	thing I want to show you before we move on
07:32	. Let me uh let's see here . Yeah ,
07:34	I guess I'll just leave it up here , just
07:36	like this . Let's do the problem again . 4/12
07:40	. Yes . All right . What we did is
07:42	we divided by two got an answer , then we
07:45	divided by two again and got an answer . But
07:47	if you notice carefully , you can actually divide by
07:49	two . Of course we did that . But actually
07:51	there's something else that divides into top and bottom here
07:54	. What if I divide by four , divide this
07:56	by four . And also this by four , divide
07:59	by four , divide bottom by four . What will
08:03	I get four divided by four is one and 12
08:06	divided by four is three . Notice I get exactly
08:09	the same answer . One third is the final answer
08:12	to this problem and I get the same answer no
08:14	matter how I simplify the problem . So here's the
08:17	bottom line when you're simplifying fractions , what you want
08:20	to do is try to divide the top and bottom
08:23	by the largest thing possible that you can do because
08:27	if you do that then you can get to the
08:29	simplified answer in only one step . But a lot
08:32	of times we don't often think of the best number
08:35	two divide by but we might notice that they're even
08:37	numbers and we can divide by two . And if
08:39	you divide by two then you get to an intermediate
08:42	answer and then you divide by two again . So
08:44	I guess my point is I don't really care what
08:45	you divide by . As long as it's legal ,
08:48	I'm okay with it . Some teachers are gonna get
08:50	mad at you . If you take more than one
08:51	step to get to the answer , I'm not going
08:53	to be like that . I don't care . I
08:54	just want you to get to the right answer .
08:56	So if you happen to see that you can divide
08:59	by four , awesome . Do it If you're not
09:01	sure about that , but you know , you can
09:02	divide by two , then just do it multiple times
09:05	. That's fine with me to to me . As
09:07	long as you get the right answer , I'm happy
09:10	. All right . Let's take a look at the
09:12	problem . 2/10 . How do we simplify that ?
09:15	We rewrite the fraction as to over 10 . What
09:17	can we divide top and bottom line ? Well ,
09:20	I can divide the top and the bottom by two
09:23	because both are divisible by 22 divided by two is
09:25	1 10 divided by two is five . So what
09:29	we're claiming is that the fraction 1/5 is exactly the
09:32	same thing as 2/10 . So what we want to
09:35	do is see if that's true , here's 1/10 here's
09:38	2/10 . And then we're asking ourselves is the fraction
09:42	1/5 here the same . And of course we can
09:44	pick it up and put it right on top .
09:46	You can see that they're not just close . They're
09:48	not just kind of sort of equal . They're exactly
09:50	the same thing . Two tents is exactly the same
09:53	thing as 1/5 . So for the rest of the
09:55	problems , we're not going to use the magnets .
09:57	But I just want you to make sure that ,
10:00	you know , that you can divide by whatever you
10:02	want . As long as it's legal . And if
10:04	you get to a place where you can still simplify
10:06	it further , then just do it again . If
10:09	it takes more than one step , then all it
10:11	means is there was probably something larger that you could
10:14	have divided by . But it's not like wrong .
10:17	If you take more than one step , I'm just
10:18	telling you there's more than one path to the answer
10:20	in math . I think I've been telling you over
10:22	and over there is almost always more than one way
10:25	to get to an answer . So when I'm doing
10:27	these problems for you , you might see a different
10:29	way in a different path to the answer and that
10:31	is okay , Let's take a look at the problem
10:35	. 9/18 . Let's simplify 9/18 . This is an
10:40	excellent example of that . All right . What can
10:43	we do ? Well , we can divide the top
10:46	and bottom by three . How do I know ?
10:48	Because I just know that this is divisible by three
10:50	and this is divisible by three . So let's do
10:52	it divide top by three , divide bottom by three
10:55	. What do I get ? Nine divided by three
10:57	is 3 18 divided by three is six . Now
11:00	I'm going to circle this answer , but then I
11:01	noticed well wait a minute , I can further simplify
11:04	this further 36 . I can divide it by three
11:07	again . Okay , I've been divided by three again
11:10	, so I get to an intermediate answer but this
11:11	is not fully simplified . I can divide again by
11:14	three . Three divided by three is 16 divided by
11:19	three is two . And so the answer is finally
11:23	one half because I cannot simplify or divide this any
11:26	further to make it any simpler . And it took
11:29	kind of like two steps to get to hear step
11:31	number one , and then for here to get to
11:34	step number two . But let's really quickly just do
11:36	the whole thing again . 9/18 Notice that it took
11:40	two steps , so there must be something even bigger
11:42	. I can divide by if I happen to see
11:44	it . What about if I divide the top by
11:46	nine and the bottom by nine . Now you might
11:48	not have seen that , but if you do you
11:50	know that nine divided by nine is 1 , 18
11:53	divided by nine is too because two times nine is
11:57	18 and you can see you get exactly the same
11:59	answer . So I guess I'm going to stop solving
12:02	all of these problems . Two ways . What I
12:04	want you to know is that I don't care if
12:06	you see the perfect thing to divide by , the
12:08	best thing to divide by here would be nine because
12:10	it only gives one step . But oftentimes I don't
12:13	really see that either , and it might take more
12:15	than one step to get there . So , as
12:17	we go forward , if it takes me a different
12:19	number of steps than you find there's more than one
12:22	way to do these problems . All right , so
12:25	let's do the next problem . What about 4/20 ?
12:29	Right . So , I'm gonna write 4/20 . I
12:33	know right away I could divide by four and then
12:35	divide by four because I know that they're both divisible
12:37	. But let's just say that I didn't notice that
12:39	I know these are even numbers . So I divide
12:41	by two and by 24 divided by two is 2
12:45	20 divided by two is 10 . But then I
12:48	noticed , oh , man , I can do it
12:50	again because these are again even number . So I
12:52	write to tents down again and then I say ,
12:55	all right , I'm going to divide the top by
12:57	two . In the bottom by 22 divided by two
13:00	is one , 10 divided by two is five .
13:04	And the answer is 1/5 . If I did not
13:08	divide by two and then by two again , what
13:11	if I noticed and started off by just dividing by
13:13	four ? Well , four divided by four is one
13:17	20 divided by four is 5 . I would go
13:20	straight from here to the answer in one step .
13:22	If I choose the best thing to divide by .
13:24	But to me , as long as you get the
13:26	right answer , that's fine . I'm totally fine with
13:28	that . All right enough of that . Let's take
13:32	a look at 12 15 . So let's simplify that
13:36	. Rewrite the fraction . Put the 12 and the
13:38	15th . Draw your longer fraction . Bar what can
13:41	I divide this by ? Well , I know I
13:44	can divide by three and by three . So put
13:47	up here , I'm gonna divided by three , divided
13:49	by 3 . 12 divided by three is four because
13:52	four times three is 12 , 15 , divided by
13:55	three is five because five times three is 15 .
13:57	So the answer is 4/5 and I can't simplify this
14:01	further because I can't divide top and bottom by something
14:04	to make that any simpler ? All right . What
14:07	about the uh problem ? 8/20 . Right . Now
14:14	, I know that there are probably other things I
14:16	can divide by but let's just say , I don't
14:18	know and I just noticed that these are even numbers
14:21	, right ? So I'll just divide this by two
14:23	because I know they're both even , that's definitely gonna
14:25	work . Eight divided by two is 4 20 divided
14:29	by two is 10 . And then I say ,
14:31	is this simplified , nope , because it's still even
14:35	numbers ? So I just write the 4/10 again and
14:37	I do the same thing again , I can divide
14:38	both by two by the top by 25 Bottom by
14:41	two , I get four divided by two is two
14:44	and 10 divided by two is five . So the
14:47	answer is to fifth . There is nothing I can
14:49	divide further to make this simpler now . Uh I
14:53	said I won't do this for every problem but I
14:55	can't resist doing it here , let's rework the problem
14:57	again . What else could I divide by ? Uh
15:00	What else is divisible here ? Actually you can divide
15:02	the top and bottom by four . You may not
15:05	see that right away , but eight divided by four
15:07	is two 20 divided by four is what ? Five
15:12	? Because five times four is 20 . So you
15:14	see you get the same answer , It's just if
15:16	I pick the bigger number , uh which is by
15:19	the way called the greatest common factor . That's why
15:21	we learned what the greatest common factor was . Because
15:23	if you always divide by the greatest common factor ,
15:25	then you'll get the simplified fraction in the fewest number
15:29	of steps . So if you happen to see that
15:31	you can divide by four , then you're gonna get
15:33	the answer in one step . But if you're like
15:34	the rest of us , who doesn't always see that
15:37	? Just divide by two ? Because it's even divide
15:39	by two again , maybe you have to do it
15:40	twice . But then you always get the right answer
15:43	with no problems . All right . Let's take a
15:47	look . Our final few problems . We'll take a
15:51	look at 14/18 . What can I divide top and
15:56	bottom by ? Well , I can see right away
15:57	that these are even numbers . So I'm going to
15:58	divide the top and the bottom five to top by
16:04	two . Bottom by two . What is 14 divided
16:07	by two ? Seven ? Because seven times two is
16:09	14 , One is 18 divided by 29 because nine
16:12	times two is 18 . The answer is seven nights
16:15	. I cannot divide top and bottom by anything further
16:17	to make this any simpler . So , I am
16:19	done . All right . Only two more problems .
16:24	What about 26 ? Right . 26 . Again .
16:30	I see that they're both even numbers . So right
16:32	. 26 on the top . 28 on the bottom
16:35	. And I'm going to divide the top and the
16:37	bottom by two . Now , these are a little
16:40	tricky because they're larger numbers . Right ? But when
16:43	you think about it , I think you can convince
16:44	yourself that 26 divided by two is 13 and 28
16:48	divided by two is 14 . And if you're not
16:52	sure about that , you can go off to the
16:54	side 26 divided by two and go through it all
16:56	and you'll get 13 and do this one , you'll
16:58	get 14 . I know we've covered all that in
17:00	the past . So at some point I have to
17:02	start Assuming that you know how to divide numbers ,
17:04	which we've done before you get 13/14 here , and
17:08	these are the final answers . I cannot simplify that
17:10	any further . All right . And then finally ,
17:14	here is our last problem . What about 24 and
17:20	96 ? So , I'll write 24 again and 96
17:24	again . Now I don't really know exactly what the
17:27	best number is to divide by , but I know
17:29	that they're both even so I'm going to divide the
17:31	top by two and I'll divide again the bottom by
17:34	two . Now you might need to go off to
17:37	the side and take 24 divide by two and so
17:41	on to get the answer . But I think you
17:42	can convince yourself that this is 12 on the top
17:44	because 12 times two is 24 And then 96 divided
17:48	by two is actually gonna work out to be 48
17:51	. You take 96 divided by two . We've done
17:53	all that many times , you get an answer of
17:55	48 . But this is not perfectly simplified because these
18:00	are again uh even numbers . So what will I
18:04	do ? I will divide the top and bottom by
18:08	two again . Right , what am I going to
18:11	get ? 12 divided by two is six and 48
18:15	divided by two . When you do the division and
18:17	work that out , you'll get 24 . These again
18:21	are even numbers . So six , here , I'm
18:27	going to be a little bit smarter . I can
18:28	divide by two again because there are even numbers but
18:31	I realize now that I can just divide this by
18:34	six , I can divide this by six and this
18:37	by six because they're both divisible by six . Six
18:41	, divided by six is 1 , 24 divided by
18:43	six is four because four times 6 24 . So
18:46	the answer is 1/4 . So notice that we had
18:50	like a multi step thing going on here , we
18:52	divided one time to get this thing and then we
18:54	divided another time to get this thing and then I
18:57	guess I'll do it like this another time to get
18:59	this . So we did one division to division three
19:01	divisions . And it always worked out but it was
19:04	multiple steps . So when you have multiple steps like
19:07	that , to simplify a fraction , all it means
19:09	is that there was a larger number , I could
19:11	divide top and bottom by Um to get exactly the
19:14	same thing . What do you think that is ?
19:17	I think I could actually probably divide the top and
19:19	bottom by 24 . What I'm claiming is if I
19:23	take the top and the bottom And divide by 24
19:29	and divide by 24 , what will I get the
19:32	answer is 1/4 because four times 24 is 96 Let's
19:35	check it . 24 times 44 times 4 , 16
19:39	carry the 12 times four is eight plus one is
19:42	nine , so four times 24 is this and one
19:45	times 24 is this ? So if I had looked
19:47	at this problem and I was a human calculator ,
19:50	which I'm not , but if I was I could
19:52	say , Oh I'll just divide top and bottom by
19:54	24 and I'll get the answer in one step ,
19:56	but nobody can do that . I mean I don't
19:57	think anybody , most people cannot do that . So
20:00	instead do it like the rest of us , we'll
20:02	divide by two and get this . Now actually we
20:04	divided by two again . But here , if you're
20:06	clever , you can see that you can divide top
20:08	and bottom by 12 , because 12 divided by 12
20:12	is 1 , 48 divided by 12 is four .
20:15	Remember for multiplication 12 times four is 48 . So
20:18	you might have been able to shortcut a couple steps
20:20	by dividing top and bottom by 12 here , but
20:22	if you didn't think of it divided by two again
20:24	, and then here we divided by six and so
20:27	on and get the final answer of 1/4 . So
20:29	simplifying fractions is really important because when we add or
20:33	subtract or multiply or divide fractions , the answer we
20:36	get , we'll always have to check if it's in
20:38	lowest terms and if it's not , we'll have to
20:40	do this process to get it into lowest terms .
20:42	Don't forget what's happening . What we're saying is 24
20:46	, of a pizza Is exactly the same amount of
20:49	food as 12 , of a pizza or 12 slices
20:52	out of 48 , which is exactly the same thing
20:55	as six slices out of 24 , which is exactly
20:58	the same thing as one slice out of four .
21:00	They all represent the same thing , but they're just
21:03	written slightly different . That's all that that means .
21:05	So I'd like you to solve all of these yourself
21:07	. Follow me on the part two , we'll wrap
21:08	up our skills with simplifying fractions .

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